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Theorem 3brtr3i 3791
 Description: Substitution of equality into both sides of a binary relation. (Contributed by NM, 11-Aug-1999.)
Hypotheses
Ref Expression
3brtr3.1 𝐴𝑅𝐵
3brtr3.2 𝐴 = 𝐶
3brtr3.3 𝐵 = 𝐷
Assertion
Ref Expression
3brtr3i 𝐶𝑅𝐷

Proof of Theorem 3brtr3i
StepHypRef Expression
1 3brtr3.2 . . 3 𝐴 = 𝐶
2 3brtr3.1 . . 3 𝐴𝑅𝐵
31, 2eqbrtrri 3785 . 2 𝐶𝑅𝐵
4 3brtr3.3 . 2 𝐵 = 𝐷
53, 4breqtri 3787 1 𝐶𝑅𝐷
 Colors of variables: wff set class Syntax hints:   = wceq 1243   class class class wbr 3764 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765 This theorem is referenced by:  iap0  8148
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